How can these textures get fixed?
By UV-unwrapping them correctly.



Without UVmap the textured object appears all made of one color (a sort of average color of the texture)



By unwrapping the geometry in proper way (in this simple case, just a basic unwrap is enough) you'll be able to se the texture cover the plane as expected.

Notice that the Uvmap in the uv/image editor has the two triangles located in a way that reflect the faces topology.



Resetting the UVmap brings all the faces to cover the whole texture..so the triangles are both in the same location, overlapped. So each faces show the same part of the mesh, leading to the behavior you step onto.

Notice how the vertex order of the faces determines an unexpected rotation of the textures on the geometry. This brings a discontinuty in the textures which now don't follow anymore the mesh. Every face has it's own UV.

You have to re-unwrap them.